The Moduli Space of Acute Triangles

TL;DR

This paper explores the moduli space of similarity classes of acute and right triangles, revealing a deep connection to elliptic curves and showing a three-to-one correspondence between these geometric objects.

Contribution

It introduces the moduli space of acute and right triangles and establishes a surjective, three-to-one mapping to the moduli space of elliptic curves, linking geometry and algebraic curves.

Findings

The moduli space of these triangles maps onto the elliptic curve moduli space.

Every elliptic curve can be associated with an acute or right triangle.

The mapping between these spaces is generically three-to-one.

Abstract

As an introduction to the concept of "moduli space" we consider the moduli space of similarity classes of acute and right triangles in the plane. This has a map to the moduli space of elliptic curves which is onto and generically three-to-one. The reason is that from any acute or right triangle we can construct an elliptic curve, and every elliptic curve is isomorphic to one constructed this way.